Solve problem 3 from Advanced Linear Algebra relatef to Matrices and eigen vectors.
3. Let A be an n x n – matrix over C. Recall that A is diagonalizable if and only if there*exists an invertible matrix B such that B- 1 AB – D where D is a diagonal matrix .Show that A is diagonalizable if and only if (" has a basis of eigenvectors of A .
Let A be an arbitrary m x n matrix, and let B be an arbitrary n x p matrix.
a. Show that any vector x that is in the column space of AB is also in the column space of A.
b. Is it also true that any vector x that is in the column space of AB is also in the column space ofB? Why or why not?
c. Is it also true that any vector x that is in the column space of A is also in the column space ofAB? Why or why not?
Let “a” be a positive n-bit binary number and let “b” be a positive m-bit binary number.(a) What is the range of values that the multiplication of those two numbers (a∗b) will have? Hint: Only specify the minimum and the maximum values that the resulting multiplication can have. Express each of these values in base 10, as a formula which is based on the parameters n and m.(b) Provide an analogous response when adding the two positive n-bit and m-bit numbers (a + b).
5.6.15. Let f : A –> B be an arbitrary function.
(a) Prove that if f is a bijection (and hence invertible), then f^-1(f(x)) = x for all x belonging to A, and f(f^-1(x)) =
x for all x belonging to B.
(b) Conversely, show that if there is a function g : B –> A, satisfying g(f(x)) = x for all x belonging to A, and
f(g(x)) = x for all x belonging to B, then f is a bijection, and f^-1 = g.
Need help with math homework problem. Please look at the image.
1. LetEm : {partitions 71 of [n] : {1, …, n}} Where a partition is an unordered set of subsets7T = Sll – – – IS;C = {$1, …,Sk} With Sl- C [n] such that each element of [n] is in exactly one of the Si. Find a formulafor ”(0, 1) for n = 4, Where u is the Mobius function of this partially ordered set7Where 0:1|—|’n,, 121…”are the smallest and largest elements, respectively. 2. Prove a formula for the Mobius function of the partially ordered setZ20 X Z20 Where((1,1)) g (ad) ifa g C, and bg d
Please help with the attached question. The question is a fill-in-the blank proof
Let E CR and F C R. Let X = {(r,y)|:r,yE IR and y = I}. Prove that ifX C E X F, thenE x F = R2 by filing out gaps in the following argument. Proof. We are given that X C _. We need to show that _ and Let (11,5) E E X F. Then, a E_ and b E_. If a. E_, then a E_ because E C_. If I; E_Tthen b E_ because F C_. Therefore, (o,b) E 13.2 because _. We showed that _. Let ((1,5) E 1R2. Then, a E_ and b E_. Therefore, by definition of X , we have (and) E Kand (b,b) E X. Since (the) E X and _, we have _. Since (b,b) E X and we have _.Therefore, (a, b) E E X F. We showed that _. _T
Let E be an n x n elementary matrix and A be an n x n matrix. Which of the following
statements are ALWAYS true? (RS=row space, NS=Nullspace, CS=column space)
(i) NS(EA) = NS(E)
(ii) RS(EA) = RS(A)
(iii) CS(EA) = CS(A)
(iv) RS(EAT ) = CS(A)
(v) NS(A) = NS(AE)
A. (ii) and (iii)
B. (i) and (ii)
C. (ii) and (iv)
D. (i), (ii) and (v)
E. None of the above
2. Let Dn be the dihedral group of symmetries of a regular n-gon, and let Cn ⊂ Dn be the subgroup of rotations. Let also H ⊂ Cn be an arbitrary subgroup. Prove that H is a normal subgroup of Dn.
Remark: H is a subgroup of Cn, so it is also a subgroup of Dn. Note that H is a normal subgroup of Cn, because Cn is Abelian. This means that ghg−1 ∈ H for any g ∈ Cn. What you need to prove is a stronger result: H is normal in Dn, meaning that ghg−1 ∈ H for any g ∈ Dn, and not only for any g ∈ Cn.
Hint: One possible approach is to show that for any element g ∈ Dn, the set gHg^-1 = {ghg^-1 | h ∈ H} is a subgroup of Cn. What is the order of this subgroup?
What do we know about subgroups of cyclic groups? Another approach is to use geometric arguments to prove that for any reflection g ∈ Dn Cn and any rotation r ∈ Cn, one has grg^-1= r^-1.
2. Let consumption equal C = 100 + 0.75Y. The equilibrium condition is that income (Y) equals planned expenditures, or Y = C + I, where I is investment.
a. Solve for equilibrium levels of income and consumption if I = 500.
b. Find reduced-form equations for Y* and C* in terms of the exogenous variable, I.
c. Describe the comparative statics results of this system with respect to changes in I.
Let C_b(R) denote the space of all bounded, continuous functions f : R → C. Let C_0(R) denote the set of continuous functions f : R → C for which lim x→±∞ f(x) = 0. How do you prove that C_b(R) and C_0(R) are complete in the uniform metric?
Hi there! Click one of our representatives below and we will get back to you as soon as possible.